A new approach to integrable evolution equations on the circle

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be exp...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2021, Vol.477 (2245), p.1-28
Main Authors: Fokas, A. S., Lenells, J.
Format: Article
Language:English
Subjects:
finite-gap solution
Fokas method
integrable evolution equation
inverse scattering
Riemann-Hilbert problem
unified transform method
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Summary:We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.
ISSN:1364-5021
1471-2946
1471-2946
DOI:10.1098/rspa.2020.0605